Pseudometrizable bornological convergence is Attouch-Wets convergence

Let S be an ideal of subsets of a metric space (X, d). A net of subsets (Aλ) of X is called S-convergent to a subset A of X if for each S ∈ S and each ε > 0, we have eventually A ∩ S ⊆ A _λ^ε and Aλ ∩ S ⊆. A ^ε. We identify necessary and sufficient conditions for this convergence to be admissible and topological on the power set of X. We show that S-convergence is compatible with a pseudometrizable topology if and only if S has a countable base and each member of S has an ε-enlargement that is again in S. Further, in the case that the ideal is a bornology, we show that S-convergence when pseudometrizable is Attouch-Wets convergence with respect to an equivalent metric.